On the spectrum of the Hodge Laplacian on sequences

Abstract: Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. Here we introduce a Hodge-theoretic approach to spectral theory and dimensionality reduction for probability distributions on sequences and simplicial complexes. We demonstrate that this Hodge theory has desirable properties with respect to natural null-models, where the underlying vertices are independent. We prove that for the case of independent vertices in simplicial complexes, the appropriate Laplacians are multiples of the identity and thus have no meaningful Fourier modes. For the null model of independent vertices in sequences, we prove that the appropriate Hodge Laplacian has an integer spectrum, and describe its eigenspaces. We also prove that the underlying cell complex of sequences has trivial reduced homology. Our results establish a foundation for developing Fourier analyses of probabilistic models, which are common in theoretical neuroscience and machine-learning.